Sure! Let's look at the expression [tex]\(x^2 + 2x + c^2\)[/tex] step by step to understand its components and make sure we comprehend it fully:
1. Identify the Terms:
- [tex]\(x^2\)[/tex]: This term is a square of [tex]\(x\)[/tex], indicating a parabolic relationship that would open upwards since the coefficient (1) is positive.
- [tex]\(2x\)[/tex]: This term is linear in [tex]\(x\)[/tex], indicating a slope.
- [tex]\(c^2\)[/tex]: This term is the square of the constant [tex]\(c\)[/tex]. Since it is squared, it is always non-negative.
2. Combine the Terms:
The expression is made up of three components combined through addition:
- [tex]\(x^2\)[/tex] contributes a parabolic shape.
- [tex]\(2x\)[/tex] adjusts the slope of the parabola.
- [tex]\(c^2\)[/tex] shifts the entire graph of the function vertically.
3. Resulting Expression:
When you combine these terms together, you retain each part of the original expression in its respective position. Each term influences the overall shape and position of the resulting function.
Thus, the given expression is:
[tex]\[x^2 + 2x + c^2\][/tex]
This is already simplified, and no further calculations or algebraic manipulations are necessary. This expression tells us that it is a quadratic expression in [tex]\(x\)[/tex] with a constant term [tex]\(c^2\)[/tex].
